In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A_{1},...,A_{n} of C have a biproduct A_{1} ⊕ ⋯ ⊕ A_{n} in C.
(Recall that a category C is preadditive if all its homsets are Abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of Abelian groups. Recall also that a biproduct in a preadditive category is both a finite product and a finite coproduct.)
Warning: The term "additive category" is sometimes applied to any preadditive category, but Wikipedia does not follow this older practice.
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Definition
A category C is additive if
 it has a zero object
 every homset Hom(A,B) has an addition, endowing it with the structure of an abelian group, and such that composition of morphisms is bilinear
 all finite biproducts exist.
Note that a category is called preadditive if just the second holds, whereas it is called semiadditive if both the first and the third hold.
Also, since the empty biproduct is a zero object in the category, we may omit the first condition. If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets.
Examples
The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given pointwise, and biproducts are given by direct sums.
More generally, every module category over a ring R is additive, and so in particular, the category of vector spaces over a field K is additive.
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